We remark that the bound of Corollary REF  is proved in [1]}, which was adapted from the work of Matoušek and Tancer [2]}. In the same paper, Matoušek and Tancer show that the gap \(\mathcal {C}(X)-\mathcal {L}(X)\)  between the collapsibility and Leray numbers could be arbitrarily large. On the other hand, we next verify that they coincide on a distinguished class of simplicial complexes.
